Combining nonlinear public goods games with population dynamics, this Demonstration plots the density of cooperators in the population as a function of the population density (1 minus the empty spaces, ). For , the population has reached the ecological limit, while for , it is extinct.

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This Demonstration combines the results of [1] and [2]; [1] discusses the impact of nonlinear benefits in public goods games, while [2] incorporated ecological dynamics into standard evolutionary game dynamics.

Here the nonlinearity is given by the synergy/discounting parameter . When , we have linearity. When , the benefits get added up synergistically, with each additional cooperator contributing more than the previous one. When , the benefits saturate for a certain number of cooperators. The frequencies of cooperators and defectors in the population are given by and . Since each individual in the population has one of these two strategies, we have . However, the sum may be less than the carrying capacity of the environment. Hence if we consider and to be the fractions of the two strategies, then we can have as the fraction of empty spaces. Tracking the ecological dynamics then amounts to looking at the temporal evolution of the population density , while the evolutionary dynamics are the temporal evolution of the strategies within the population density. The dynamics of the empty sites and the fraction of cooperators are given by

,

,

where ;

is the benefit multiplier; is the death rate; is the synergy/discounting factor, and is the group size.